3. FRIEDMANN OSCILLATIONS: THE RISE AND FALL OF DOMINANT
COMPONENTS

where
Rµ is
the Ricci tensor,
is the Ricci scalar,
gµ is
the metric tensor describing the local curvature of space (intervals of
spacetime are described by ds2 = gµdxµdx),
Tµ is
the stress-energy tensor and
is the
cosmological constant. Taking the
(µ, ) = (0, 0)
terms of Eq. 10
and making the identifications of the metric tensor with the terms in
the FRW metric of Eq. 1, yields the Friedmann Equation:

(11)

where R is the scale factor of the Universe,
H = /
R is Hubble's constant,
is the density
of the Universe in relativistic or non-relativistic matter, k is
the constant from Eq. 1 and
is the
cosmological constant.
In words: the expansion (H) is controlled by the density
(), the
geometry (k) and the cosmological
constant ().
Dividing through by H2 yields

If we are interested in only post-inflationary expansion in the
radiation- or matter-dominated epochs we can ignore the
term and multiply
Eq. 11 by 3 / 8G to get

(15)

which can be rearranged to give

(16)

A more heuristic Newtonian analysis can also be used to derive Eqs. 14
& 16 (e.g.
Wright 2003).
Consider a spherical shell of radius R expanding at a velocity
v = HR, in a universe of density
.
Energy conservation requires,

(17)

By setting the total energy equal to zero we obtain a critical density
at which v = HR is the escape velocity,

(18)

However, by requiring only energy conservation (2E =
constant not necessarily E = 0) in Eq. 17, we find,